taken, and its attraction upon the plumb line accurately computed by Dr Hutton, who found that the deviation was about half what it ought to have been, if the density of the mountain had been equal to that of the earth. Hence it was inferred, that the mean density of the earth must be nearly double the density of that mountain, or about five times the density of water. Nearly the same result was obtained from the experiments of Cavendish. This is also in some degree confirmed by the perfect stability, or tendency to return to a state of repose, in the waters of the ocean when disturbed. For it has been proved by Laplace, from the theory of gravity, that if the mean density of the earth was equal to, or less than, that of the waters of the ocean, the equilibrium would be unstable, and if any cause disturbed it, the tendency of gravity would be to increase the motion, and make the sea overflow its shores, and destroy the present system of the earth. In calculating the oblateness of the earth, Newton did not take into consideration the equilibrium of the fluid upon the surface, but contented himself with the form that would result from the supposition, that the pressure of the fluid at the centre of the earth, in two small canals, drawn from that centre to the surface of the earth at the equator, and at the pole, would exactly balance each other. Bouguer showed that these canals might be in equilibrium, and yet the fluid be unstable at the surface of the earth; and he showed that it was also necessary that the force, acting on any point at the surface, should be perpendicular to that surface. Clairaut, in his theory of the earth, published in 1743, proved that there might be cases where both these conditions were satisfied, and yet the fluid be unstable, and that, for a permanent equilibrium, it was necessary that the fluid in any canal, taken at pleasure, should be in equilibrium. He investigated in that work the analytical expression of this principle, supposing the earth to be formed of concentric ellipsoidal strata, couches de niveau, increasing in density from the surface to the centre. He also discovered a curious theorem, by which the increase of gravity in going from the equator to the pole, determined by the length of a pendulum vibrating in one second of time in different latitudes, was connected with the oblateness of the earth; the fraction denoting the increment of gravity being as much above, as the fraction denoting the oblateness is less than; so that the sum of these two quantities would be, for all probable suppositions of the densities of the strata of the earth. D'Alembert, who wrote on this subject several times, was the first who calculated the attraction of spheroids, whose meridians were not elliptical. Legendre, by an ingenious method, making use of the properties of a singular species of functions, took into consideration the case where the meridians differed from an elliptical form, and varied for different longitudes. This was also done by Laplace, who devotes the third book of his Mécanique Céleste almost exclusively to the theory of the earth, using functions somewhat similar to Legendre's, and founding his calculations upon a remarkable equation of partial differentials, discovered by him, by which the attraction of a spheroid, upon a point situated upon its surface, can be obtained without any integration. This equation is generally correct, but there are cases where it might fail, like Taylor's theorem, and almost all other theorems of a like general nature, when applied to some peculiar cases, in a different manner from what was usually intended. This was the case with the exception in Laplace's equation, mentioned by Lagrange, in computing the attraction of a spherical shell upon a point, situated upon its surface. The same defect was also pointed out by Mr Ivory, who had solved the general problem, by a direct process of integration, in his usual elegant manner. He also proved, that Laplace had made some deductions from his formula, which were not absolutely warrantable; but the manner in which the subject was treated by Mr Ivory evinced, in some degree, a disposition to speak too slightly of Laplace's method. These strictures induced Laplace again to bring forward his demonstration, in the paper mentioned at the beginning of this review, read to the Royal Academy of Arts and Sciences in 1818, in which, without condescending to mention Mr Ivory by name, he says, 'Quelques géomètres ne l'ayant pas bien saisie l'ont jugée inexacte;' and he then goes on to show how the difficulty in question would have been avoided, if they had restricted his equation to the cases for which it had been designed. In an additional memoir published in the Transactions for 1818, he supposes the density of the strata of the earth to increase with the pressure of the superincumbent mass, according to the suggestion of Dr Young, and he assumes as a probable hypothesis, that the ratio of the increments of the pressure and density are proportional to the density, instead of being constant as in the gaseous fluids, supposing the solid matter of the earth to resist the increase of density more powerfully than in the ratio, which prevails in the gases. This hypothesis makes the oblateness, and satisfies all the known phenomena, depending on the law of the densities of the strata, namely, the variation of the degrees of the meridian and of gravity, the precession of the equinoxes, the nutation of the earth's axis, the lunar equations, depending on the oblateness of the earth, and the ratio of the mean density of the earth to that of water, which was found, by the experiments of Cavendish, to be 51⁄2, and, in this hypothesis, the density at the centre of the earth would be about twelve times that of water, being greater than that of lead. In this calculation, Laplace supposes the temperature of the earth to be uniform throughout the whole mass. He, however, observes that it was possible, that the heat might be greater towards the centre than at the surface, as would necessarily be the case if the earth at any period had been much heated, and was gradually cooling, conformably to his ideas of the origin of the present arrangement of the solar system, given in the last edition of his Exposition du Système du Monde. He discusses this point, and proves from astronomical phenomena, that this decrease has been insensible since the time of Hipparchus. His reasoning is in substance as follows. If the temperature of the earth was suddenly to decrease one degree of Fahrenheit's thermometer, its dimensions would be decreased by a quantity which, for the sake of argument, may be supposed a two hundred thousandth part, which is nearly what takes place in glass. In consequence of this, the angular velocity of rotation would be increased about one hundred thousandth part, because, by the principle of areas, the sum of the areas described by each particle of the earth about its axis of rotation, would not be Or, in other words, the momentum of rotation found by multiplying each particle by the square of its distance from the axis of rotation, and by its angular velocity, would be the same for the whole mass before and after the change of temperature. altered by this change of temperature, and, as the length of the day is 86,400 seconds, this length would, by this means, be decreased nearly one second. This change of dimensions would not affect the earth's mass, or its attraction on the moon, and the absolute time of the moon's periodical revolution, which is nearly 27 days, would not be altered; but, being measured by days, which have decreased in length nearly one second, that period would appear longer by 27 times that decrement, or 23 seconds; so that if the earth's temperature had decreased one degree since the time of Hipparchus, the moon's periodical revolution about the earth would appear to have increased 23 seconds. Now it has been found, by observation, after allowing for the acceleration of the moon's motion, arising from the secular change in the eccentricity of the earth's orbit, that the periodic time of revolution has not suffered any perceptible change, since the earliest observations on record; therefore no change of temperature, of any moment, could have taken place in the earth during that period. The theory, combined with astronomical observations, has done more for the determination of the figure of the earth, than the actual measures of the degrees of the meridian, which have been made in several countries, with great labor and expense, but without obtaining that degree of accuracy, which was reasonably to have been expected. The degree measured in Lapland, by Maupertuis and his associates, has been found more than two hundred toises too great, by the late measurement of Svanberg.* The degree of Austria, by Liesganig, has been proved by Baron de Zach to be so very inaccurate, as to be wholly undeserving of notice. measures at the Cape of Good Hope, Peru, and Pennsylvania, are considered tolerably accurate, but not of the first order. The late ones in France and England have The *The discovery of this mistake would have mortified extremely the vanity of Maupertuis, who, upon his return from this northern expedition, published immediately an account of it, without waiting to know the result of the operations in Peru, wishing to appropriate to himself the whole honor of the operations, to which, in fact, he had contributed but a small portion. He also caused a portrait of himself to be engraved, in a Lapland dress, with his hand resting upon the northern part of a terrestrial globe, as if he was compressing it, and for some time he was called by his countrymen, 'l'aplatisseur de la terre;' (the flattener of the earth;) instead of giving the surpassed all others in the accuracy of the instruments, and precautions of the observers; but even these, particularly the English measurement, have not escaped animadversion, on account of their discrepancies. The most probable combination of these measures, shows that the oblateness of the earth is between and, agreeing with the results of the lunar theory. It may also be observed, that this oblateness being less than, proves by Clairaut's theorem, beforementioned, that the earth increases in density from the surface towards the centre, confirming the proof deduced before from other sources. The precession of the equinoxes is intimately connected with the theory of the earth, and the oblateness of its form. Newton was the first who discovered its cause, and that, in his hypothesis of a homogeneous earth, it was produced by the attraction of the sun and moon upon the protuberant matter or excess above a sphere, supposed to be described about the polar diameter. The calculation of the precession, by the theory of gravity, is one of the most difficult of all the celestial phenomena, and the one which has been the most fruitful in mistakes. Newton's calculations for a homogeneous ellipsoid, in the Principia, contained important errors in principles and in data. These remained without detection till the year 1749, above sixty years after its publication, when D'Alembert first gave the true principles of solution in his Recherches sur la Précession des Equinoxes.' The general results of his solution have been confirmed by the calculations of Euler, Lagrange, and Laplace, and are now universally admitted to be true. D'Alembert proved in this work, that the sun's attraction would produce double the precession, which Newton had calculated, and that this mistake glory to Newton, who had proved forty years before, from the theory, that it must be flattened. Voltaire, who was then the friend of Maupertuis, wrote the four following lines, placed at the bottom of this portrait. 'Ce globe mal connu, qu'il a su mesurer Devient un monument où sa gloire se fonde; De lui plaire et de l'éclairer.' Voltaire, at a subsequent period, when addressing himself to the members of the Academy who composed the northern expedition, says with more justice, 'Vous avez récherché, dans ces lieux plein d'ennui, |